One at a time, put the chicken breasts between 2 sheets of plastic wrap and flatten to an even 1/2-inch thickness with a meat mallet. Sprinkle on both sides with salt and black pepper. Set aside.
Add the all-purpose flour to a baking dish. In another baking dish, stir together the almond flour, paprika, garlic powder, onion powder and 1/2 teaspoon salt. In a small bowl, whisk together the eggs with 2 tablespoons water. Dredge both sides of the chicken breasts first in the AP flour, then the egg mixture and finally the almond flour mixture. Place the coated chicken back on the baking sheet.
Heat a large ovenproof skillet over medium-high heat. (Depending on the size of the breasts, you may need two skillets.) Add the butter and 2 tablespoons oil to the skillet. Add the garlic and rosemary and cook to infuse the oil, about 1 minute. Remove the rosemary and garlic from the skillet. Make sure the oil is hot enough to be quivering in the skillet, then add the chicken by placing it in and releasing it away from you. Shake the skillet to ensure it doesn't stick. (You can add the rosemary sprig back to the skillet if you want to infuse more flavor.) Cook until the chicken begins to brown, 3 to 4 minutes, then flip the chicken and cook until the other side is browned, another 3 to 4 minutes. Remove the chicken from the skillet and drain on a half-sheet tray fitted with a rack.
In a medium bowl, whisk together the sour cream, vinegar, honey, poppy seeds and 2 tablespoons water. Set aside. Season the arugula with the remaining 2 tablespoons olive oil and salt and pepper to taste and toss. Add the apple slices and toss to coat. Plate the chicken and serve with the arugula salad. Drizzle the salad with the poppy seed sour cream. Serve immediately.